A209442 G.f. satisfies: x = Sum_{n>=1} 1/A(x)^(6*n) * Product_{k=1..n} (1 - 1/A(x)^k).

1, 1, 5, 45, 470, 5365, 64766, 813012, 10505163, 138800397, 1866712401, 25470265992, 351717013269, 4906153922941, 69030042202001, 978531875343171, 13961726654230994, 200351151383453293, 2889692388200640136, 41867983817065377259, 609091785100828769195

Offset: 0

Author

Paul D. Hanna, Apr 08 2012

Keywords

• nonn

Comments

Compare the g.f. to the identity:

G(x) = Sum_{n>=0} 1/G(x)^n * Product_{k=1..n} (1 - 1/G(x)^k)

which holds for all power series G(x) such that G(0)=1.

Examples

			G.f.: A(x) = 1 + x + 5*x^2 + 45*x^3 + 470*x^4 + 5365*x^5 + 64766*x^6 +...
The g.f. satisfies:
x = (A(x)-1)/A(x)^7 + (A(x)-1)*(A(x)^2-1)/A(x)^15 + (A(x)-1)*(A(x)^2-1)*(A(x)^3-1)/A(x)^24 + (A(x)-1)*(A(x)^2-1)*(A(x)^3-1)*(A(x)^4-1)/A(x)^34 + (A(x)-1)*(A(x)^2-1)*(A(x)^3-1)*(A(x)^4-1)*(A(x)^5-1)/A(x)^45 +...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..835

Crossrefs

Cf. A001002, A181997, A181998, A209441.

Programs

Mathematica

nmax = 20; aa = ConstantArray[0,nmax]; aa[[1]] = 1; Do[AGF = 1+Sum[aa[[n]]*x^n,{n,1,j-1}]+koef*x^j; sol=Solve[SeriesCoefficient[Sum[Product[(1-1/AGF^m)/AGF^6,{m,1,k}],{k,1,j}],{x,0,j}]==0,koef][[1]]; aa[[j]]=koef/.sol[[1]],{j,2,nmax}]; Flatten[{1,aa}] (* Vaclav Kotesovec, Dec 01 2014 *)
CoefficientList[1+InverseSeries[Series[x - 5*x^2 + 5*x^3 + 30*x^4 - 65*x^5 - 191*x^6 + 378*x^7 + 1557*x^8 + 103*x^9 - 8551*x^10 - 23911*x^11 - 37958*x^12 - 41831*x^13 - 34156*x^14 - 21179*x^15 - 10015*x^16 - 3571*x^17 - 933*x^18 - 169*x^19 - 19*x^20 - x^21, {x, 0, 20}], x],x] (* Vaclav Kotesovec, Dec 01 2014 *)

PARI

{a(n)=if(n<0, 0, polcoeff(1 + serreverse(x - 5*x^2 + 5*x^3 + 30*x^4 - 65*x^5 - 191*x^6 + 378*x^7 + 1557*x^8 + 103*x^9 - 8551*x^10 - 23911*x^11 - 37958*x^12 - 41831*x^13 - 34156*x^14 - 21179*x^15 - 10015*x^16 - 3571*x^17 - 933*x^18 - 169*x^19 - 19*x^20 - x^21 +x^2*O(x^n)), n))}

PARI

{a(n)=local(A=[1, 1]); for(i=1, n, A=concat(A, 0); A[#A]=-polcoeff(sum(m=1, #A, 1/Ser(A)^(6*m)*prod(k=1, m, 1-1/Ser(A)^k)), #A-1)); A[n+1]}
for(n=0, 25, print1(a(n), ", "))

Formula

G.f. satisfies: 1+x = A(y) where y = x - 5*x^2 + 5*x^3 + 30*x^4 - 65*x^5 - 191*x^6 + 378*x^7 + 1557*x^8 + 103*x^9 - 8551*x^10 - 23911*x^11 - 37958*x^12 - 41831*x^13 - 34156*x^14 - 21179*x^15 - 10015*x^16 - 3571*x^17 - 933*x^18 - 169*x^19 - 19*x^20 - x^21.

G.f. satisfies: x = Sum_{n>=1} 1/A(x)^(n*(n+13)/2) * Product_{k=1..n} (A(x)^k - 1).